This puzzle seems rather difficult to solve by trying random moves that appear to get one closer to the solution. However, consider the board a sequence of integers, each being the number of pennies at that location, and the moves as adding or subtracting (+1, 1, +1) to three consecutive values. Then by inspection we can come up with a solution, not worrying about whether the number of pennies is always positive, by adding the columns:
0 0 0 0 0 0 +1 (start)
0 0 0 0 1 +1 1
0 0 0 1 +1 1 0
0 +1 1 +1 0 0 0
+1 1 +1 0 0 0 0
+1 0 0 0 0 0 0 (end)
This gives you the four critical moves that must be in any solution. However, applying them directly is illegal. [But you can] just [keep] splitting the left penny (7 times), then [do the] “critical 4 moves” and you come up with the mirror image position, so you now combine to the final position. Ergo,
0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 1 0 1
0 0 0 0 0 1 0 1 1
0 0 0 0 1 0 1 1 1
0 0 0 1 0 1 1 1 1
0 0 1 0 1 1 1 1 1
0 1 0 1 1 1 1 1 1
1 0 1 1 1 1 1 1 1
1 1 0 2 1 1 1 1 1 lt +1 1 +1
1 1 1 1 2 1 1 1 1 lt +1 1 +1
1 1 1 1 1 2 0 1 1
1 1 1 1 1 1 1 0 1 lt 1 +1 1
1 1 1 1 1 1 0 1 0 lt 1 +1 1
1 1 1 1 1 0 1 0 0
1 1 1 1 0 1 0 0 0
1 1 1 0 1 0 0 0 0
1 1 0 1 0 0 0 0 0
1 0 1 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0